Stimulated Raman adiabatic passage in physics, chemistry, and beyond

The technique of stimulated Raman adiabatic passage (STIRAP), which allows efficient and selective population transfer between quantum states without suffering loss due to spontaneous emission, was introduced in 1990 by Gaubatz et al.. Since then STIRAP has emerged as an enabling methodology with widespread successful applications in many fields of physics, chemistry, and beyond. This article reviews the many applications of STIRAP emphasizing the developments since 2001, the time when the last major review on the topic was written (Vitanov, Fleischhauer et al.). A brief introduction into the theory of STIRAP and the early applications for population transfer within three-level systems is followed by the discussion of several extensions to multilevel systems, including multistate chains and tripod systems. The main emphasis is on the wide range of applications in atomic and molecular physics (including atom optics, cavity quantum electrodynamics, formation of ultracold molecules, etc.), quantum information (including single- and two-qubit gates, entangled-state preparation, etc.), solid-state physics (including processes in doped crystals, nitrogen-vacancy centers, superconducting circuits, semiconductor quantum dots and wells), and even some applications in classical physics (including waveguide optics, polarization optics, frequency conversion, etc.). Promising new prospects for STIRAP are also presented (including processes in optomechanics, precision experiments, detection of parity violation in molecules, spectroscopy of core-nonpenetrating Rydberg states, population transfer with x-ray pulses, etc.).

Figure 1

(a) The $\mathrm{\Lambda }$ linkage pattern showing $P$-field and $S$-field linkages, the single-photon detuning $\mathrm{\Delta }\equiv {\mathrm{\Delta }}_P$, and the two-photon detuning $\delta \equiv {\mathrm{\Delta }}_P-{\mathrm{\Delta }}_S$. (b) The ladder linkage pattern showing the two-photon detuning $\delta \equiv {\mathrm{\Delta }}_P+{\mathrm{\Delta }}_S$. For STIRAP it is necessary that $\delta =0$. The relative ordering of energies $E_2$ and $E_3$ does not matter when using the customary rotating-wave approximation (RWA).

Figure 2

Example of STIRAP induced by Gaussian pulses of equal peak value ${\mathrm{\Omega }}_0$ and single-photon resonance ${\mathrm{\Delta }}_P={\mathrm{\Delta }}_S=0$: (a) time dependences of the $P$ and $S$ Rabi frequencies ${\mathrm{\Omega }}_P(t)$ and ${\mathrm{\Omega }}_S(t)$; (b) adiabatic eigenfrequencies ${ϵ}_{-}(t)$, ${ϵ}_0(t)$, and ${ϵ}_{+}(t)$; (c) mixing angle $\vartheta (t)$; and (d) populations $P_n(t)$ ($n=1$, 2, 3). Dotted vertical lines separate the five phases of STIRAP discussed in this section. Adapted from [548].

Figure 3

Top: Energy levels of ${\mathrm{Ne}}^{*}$ used in the experiment by [502]. Bottom: Schematic of the experimental setup of crossed molecular and laser beams, showing the discharge source, a collimating aperture, the $P$ and $S$ laser beams, and channeltron photon detectors $D1$ and $D2$. Adapted from [39].

Figure 4

Final populations vs $P$-field detuning ${\mathrm{\Delta }}_P$ in the ${\mathrm{Ne}}^{*}$ experiment. (a) Target state population $P_3$. (b) Intermediate excited-state population $P_2$. The peak in $P_3$ and the dip in $P_2$ are typical (and mandatory) signatures of STIRAP. Adapted from [333].

Figure 5

Population of the target state 3 vs the pulse delay in ${\mathrm{Ne}}^{*}$ experiments. The large population efficiency on the left side of the vertical line is a signature of STIRAP. Adapted from [39].

Figure 6

(a) Spectral line shape analysis typical of stabilized diode lasers. (b) The STIRAP efficiency loss vs the maximum Rabi frequency accounting for broadband noise ($N$), nonadiabaticity ($A$), and both ($N+A$). From [588].

Figure 7

STIRAP in ${}^{14}\mathrm{N}{}^{16}\mathrm{O}$ molecules. Top: Hyperfine structure of the rovibrational states $X{{}^2\mathrm{\Pi }}_{1/2}(v=0,J=1/2)$, $X{{}^2\mathrm{\Pi }}_{1/2}(v=6,J=1/2)$ and $A{}^2\mathrm{\Sigma }(v=0,J=1/2)$, and linkage patterns for linearly polarized pump and Stokes fields. The numbers on the arrows indicate the relative coupling strengths of the transitions. Adapted from [39]. Bottom: Population transfer efficiency vs pulse delay. Adapted from [453].

Figure 8

Top: Numerical simulation of $P_3=0.5$ contour lines for ${\mathrm{\Omega }}_{\max }=10/T$, $15/T$, $20/T$, and $25/T$ for Gaussian pulses with delay $\tau =1.2T$. Bottom: STIRAP efficiency vs the $P$ and $S$ detunings ${\mathrm{\Delta }}_P$ and ${\mathrm{\Delta }}_S$ in an experiment with ${\mathrm{Ne}}^{*}$. Adapted from [333].

Figure 9

Time evolution of the eigenvalues and the populations in a three-state system with nonzero two-photon detuning $\delta$. The path from state 1 to state 3 passes through an avoided crossing. Adapted from [548].

Figure 10

Top: Numerical simulation of STIRAP efficiency vs the two-photon detuning for equal (dotted) and unequal (by a factor of 3) $P$ and $S$ peak Rabi frequencies. In the absence of population loss, the asymmetry just shifts the two-photon profile from resonance (solid). The irreversible population from state 2 with rate $\mathrm{\Gamma }=5/T$ erodes the off-resonance part and produces a triangular-shaped profile (dashed). Gaussian pulse shapes are assumed of width $T$. Bottom: STIRAP efficiency vs two-photon detuning in the ${}^{40}{\mathrm{Ca}}^{+}$ experiment. The magnitudes of the two fields differ by a factor of 2.5. From [484].

Figure 11

Numerically simulated contour plots of population transfer in STIRAP vs the pulse delay $\tau$ (positive for STIRAP) and the peak Rabi frequency ${\mathrm{\Omega }}_{\max }$. Solid, dashed, and dotted contours show the $P_3=0.9$, 0.5, and 0.1 values, respectively. We assumed resonant Gaussian pulses of equal magnitude ${\mathrm{\Omega }}_{\max }$ and width $T$. Left frames: no loss, $\mathrm{\Gamma }=0$. Right frames: population loss from state 2 with rate $\mathrm{\Gamma }=3/T$. Top frames: single-photon resonance $\mathrm{\Delta }=0$. Bottom frames: $\mathrm{\Delta }=10/T$.

Figure 12

Time dependences of (a) the $P$ and $S$ Rabi frequencies and (b) the populations in fractional STIRAP. As in STIRAP, the $S$ pulse arrives before the $P$ pulse, but here the two pulses vanish simultaneously while maintaining a fixed ratio. Consequently, instead of complete population transfer $1\to 3$, a coherent superposition of states 1 and 3 is created. Adapted from [558].

Figure 13

DDP-optimized STIRAP by [537]. Top: pulse shapes. Dotted curves: Gaussian pulses, dashed curves (fully overlapping with the solid curve in the pulse overlap region): ${\mathrm{\Omega }}_P(t)={\mathrm{\Omega }}_{\max }\sin [\pi f(t)/2]$ and ${\mathrm{\Omega }}_S(t)={\mathrm{\Omega }}_{\max }\cos [\pi f(t)/2]$, solid curves: ${\mathrm{\Omega }}_P(t)=g(t){\mathrm{\Omega }}_{\max }\sin [\pi f(t)/2]$ and ${\mathrm{\Omega }}_S(t)=g(t){\mathrm{\Omega }}_{\max }\cos [\pi f(t)/2]$, with $f(t)=1/(1+e^{-4t/T})$ and $g(t)=e^{-(t/2T{)}^6}$. Bottom: Population transfer error in STIRAP for the pulse shapes in the top frame. Adapted from [537].

Figure 14

Composite STIRAP. Top: The population is transferred from state 1 to 3 via a sequence of three $SP$ pulse pairs with appropriate relative phases. When there is one-photon resonance ($\mathrm{\Delta }=0$), the ordering of the two pulse constituents is reversed from pair to pair, while for nonzero $\mathrm{\Delta }$ the ordering is the same for all pulse pairs. Bottom: Final population $P_3$ vs the pulse delay and the peak Rabi frequency for a single resonant STIRAP (left) and a sequence of five resonant pulse pairs (right) with sine-squared shapes of duration $T$ and phases $({\varphi }_{P,k},{\varphi }_{S,k})$ given by $(0,4\pi /5)(\pi ,8\pi /5)(3\pi /5,3\pi /5)(8\pi /5,\pi )(4\pi /5,0)$. From [512].

Figure 15

Sequences of coincident (left) and noncoincident (right) $S$ and $P$ pulse pairs used to produce piecewise adiabatic passage (PAP). Adapted from [457].

Figure 16

Left: Linkage pattern for $N$-state chain STIRAP. Right: Effective set of parallel $\mathrm{\Lambda }$ systems after diagonalization of the subsystem of middle states. Multistate STIRAP is optimized by tuning to a dressed middle state. Adapted from [548].

Figure 17

Top: Letter-M linkage pattern for five-state STIRAP between the magnetic sublevels of two degenerate levels, ground and excited, with angular momenta $J_g=J_e=2$. The transitions are driven by a pair of laser pulses with right (${\sigma }^{+}$) and left (${\sigma }^{-}$) polarizations. Adapted from [548]. Bottom: Pulse shapes and populations for five-state STIRAP between the magnetic sublevels (indicated by numbers) in the top panel.

Figure 18

Numerically calculated transfer efficiency for chain STIRAP with $N=4$ states vs the two middle-state detunings ${\mathrm{\Delta }}_2$ and ${\mathrm{\Delta }}_3$. The $P$ and $S$ pulse shapes are Gaussian, with peak Rabi frequency ${\mathrm{\Omega }}_0$, pulse width $T$, and delay $\tau =T$, while the Rabi frequency of the pulse coupling the middle states 2 and 3 is constant and equal to $3{\mathrm{\Omega }}_0$. Left frame: ${\mathrm{\Omega }}_0=20/T$; right frame: ${\mathrm{\Omega }}_0=160/T$. The solid curves show the dressed-state resonances, i.e., the energies of the dressed middle states. Adapted from [551].

Figure 19

Straddle STIRAP in a chain with five states. Top: Shapes of the $P$, $S$, and middle ($M$) pulses. All middle-state detunings are zero. Middle: Weak middle couplings ${\mathrm{\Omega }}_{2,3}^0={\mathrm{\Omega }}_{3,4}^0=0.2{\mathrm{\Omega }}_0$. Bottom: Strong middle couplings ${\mathrm{\Omega }}_{2,3}^0={\mathrm{\Omega }}_{3,4}^0=2{\mathrm{\Omega }}_0$. Here ${\mathrm{\Omega }}_0$ is the peak value of the $P$ and $S$ fields.

Figure 20

(a)–(c) Linkage patterns between the magnetic sublevels of the levels ${}^3{P}_0$, ${}^3{P}_1$, and ${}^3{P}_2$ in ${\mathrm{Ne}}^{*}$ for various choices of $P$ and $S$ polarizations with respect to the direction of the magnetic field $B$. Adapted from [39]. (d), (e) Population transfer efficiency in ${\mathrm{Ne}}^{*}$ vs the detuning of the $P$ laser field for a set of values of the magnetic field $B$ and the polarization choice of panel (c). The $S$ laser frequency is held fixed on resonance in (d) and off resonance by 200 MHz in (e). The inset in (d) shows the narrow avoided crossing appearing in the eigenenergies for moderate magnetic fields, which blocks the adiabatic path between the initial and target states. ${\mathrm{\Delta }}_m$ is the Zeeman splitting. Adapted from [333].

Figure 21

Population transfer in ${\mathrm{Ne}}^{*}$ vs $P$ detuning and laser power for the polarization choice of Fig. 20. From [333].

Figure 22

The 450 lowest $J=0$ states of the HCN-HNC system used by [233] for studying STIRAP-like population transfer. The horizontal grouping represents a coarse-grained partitioning according to the localization of the states: group A collects levels in the HCN well, group B collects those in the HNC well, and group C are high-lying states. From [233].

Figure 23

(a) The tripod linkage augments the $P$ and $S$ fields of the $1\leftrightarrow 2\leftrightarrow 3$ $\mathrm{\Lambda }$ linkage of STIRAP by an additional state 4 coupled to the middle state 2 by a control pulse $C$. (b) The transformed tripod system in the adiabatic basis has two bright states $B_1$ and $B_2$ unconnected with two degenerate dark states $D_1$ and $D_2$, one of which is initially populated. In the adiabatic limit all couplings can be neglected except the one between the two dark states.

Figure 24

Momentum distribution for various possible orderings for the $P$ ($\pi$), $S$ (${\sigma }^{+}$), and $C$ (${\sigma }^{-}$) pulses in tripod STIRAP. Adapted from [503].

Figure 25

Formal equivalence of (top) linkages and (bottom) pulses for STIRAP in a (left) three-state system and (right) two-state STIRAP. Adapted from [550].

Figure 26

Experimental demonstration of two-state STIRAP with a ${}^{40}{\mathrm{Ca}}^{+}$ ion in a spherical Paul trap. Left-hand frames show pulses: (a) Two Gaussian pulses for Rabi frequencies as used in STIRAP. ${\mathrm{\Omega }}_1$ is the $S$ field and ${\mathrm{\Omega }}_2$ is the $P$ field. (b) Two Gaussian pulses for use with detuning ${\mathrm{\Delta }}_{\mathrm{CHP}}$ and Rabi frequency $\mathrm{\Omega }$ of two-state STIRAP. (c) The linearly chirped detuning and partial-Gaussian pulse used in the experiment. Right-hand frame: the observed transition probability. Two-state STIRAP creates an equal superposition of states $4S_{1/2}$ and $3D_{5/2}$. From [582].

Figure 27

Energy levels for the demonstration of STIRAP via a continuum in ${\mathrm{He}}^{*}$ atoms. The $P$ and $S$ lasers couple the initial state $2s{{}^{1}S}_0$ and the target state $4s{{}^{1}S}_0$ to the same ionization continuum. From [396].

Figure 28

Left: Linkage scheme for STIRAP-like population transfer into a continuum enhanced by an ancillary state $a$—an example of LICS. Adapted from [422]. Right: Linkage structure for steering population flow through lossy states. Adapted from [155].

Figure 29

Creation and measurement of a coherent superposition of magnetic sublevels in ${\mathrm{Ne}}^{*}$ atoms: beams geometry (top) and signal (bottom). The linear polarization direction of the Stokes laser is rotated by an angle $\chi$ with respect to the $x$ axis. The polarization direction of the filter laser used for detection is rotated by an angle $\alpha$. From [540].

Figure 30

(Top) Level scheme and (bottom) signal for creation and measurement of a coherent superposition of the magnetic sublevels $M=-2$, 0, and $+2$ of the metastable level ${{}^{3}P}_2$ in ${\mathrm{Ne}}^{*}$ atoms. From [204].

Figure 31

Experimental setup for the ${}^{20}{\mathrm{Ne}}^{*}$ atomic mirror of [502]. Adapted from [39].

Figure 32

Deflection of a beam of ${}^{20}{\mathrm{Ne}}^{*}$ atoms due to transfer of eight photon momenta after double adiabatic passage from the $M=2$ sublevel to $M=-2$ and back to $M=2$. The narrow undeflected original distribution is observed due to the presence of ${}^{22}\mathrm{Ne}$ isotope atoms that are insensitive to the light. The width of the deflected beam is broader because the deflection angle depends on the velocity of the particles. From [502].

Figure 33

Top: Scheme for an atomic interferometer using STIRAP with ${}^{133}\mathrm{Cs}$ atoms. The double arrows show the propagation axis of the light and the solid lines are the atomic trajectories. Bottom: Interference fringes in the population of the $6S_{1/2}$, $F=3$, $M_F=0$ sublevel vs the Raman frequency difference of the $P$ and $S$ pulses. Adapted from [572].

Figure 34

Momentum transfer in subrecoil laser cooling peaks in a ${\mathrm{He}}^{*}$ experiment. (a), (b) 1D laser cooling. (c), (d) 2D laser cooling. From [280].

Figure 35

Larmor velocity filter: experimental setup (top) and variation of the flux of deflected ${\mathrm{Ne}}^{*}$ atoms with the magnetic field (bottom). From [502].

Figure 36

(a) Scheme for vacuum STIRAP. The labels $g$, $e$, and $x$ refer to atomic levels and 0 and 1 refer to the cavity photon number. The initial state is $|e,0⟩$ and the final state, producing the outgoing photon, is $|g,1⟩$. The pulsed $P$ field is $\mathrm{\Omega }(t)$, and the $S$ field is the vacuum coupling $2g$. (b) Experimental setup used for the demonstration of vacuum STIRAP. (c) Number of counted photons vs $P$-field detuning for different cavity detunings ${\mathrm{\Delta }}_C$ (corresponding to ${\mathrm{\Delta }}_S$). Maximum at ${\mathrm{\Delta }}_P={\mathrm{\Delta }}_C$ signals vacuum STIRAP. From [538] and [209].

Figure 37

Top: An atom-cavity system emits unpolarized single photons via vacuum STIRAP. A photon traveling along path A is delayed so that it impinges on the beam splitter simultaneously with a subsequent photon traveling along path B. Bottom: Number of coinciding photodetections in the two output ports vs the time difference between the detections. When only a single path is open antibunching is observed due to Hanbury-Brown–Twiss interference (solid curve). When both paths are open but have perpendicular polarizations no interference occurs and the beam splitter randomly directs the photons to the photodiodes (dashed line). From [297].

Figure 38

(a) Control-laser-field shape, which produces a photon shape (b) mimicking the Tower Bridge in London. Adapted from [368].

Figure 39

Top: Level scheme used for STIRAP in ${}^{40}{\mathrm{Ca}}^{+}$ ions. Population is transferred between the sublevels of the metastable levels $3D_{3/2}$ and $3D_{5/2}$ via the excited sublevels of $4P_{3/2}$. Bottom: Experimental setup of ${}^{40}{\mathrm{Ca}}^{+}$ ions trapped in a segmented linear Paul trap. The detection is performed via light-induced fluorescence collected by a CCD camera and a photomultiplying tube (PMT). Adapted from [484].

Figure 40

Left: Population transfer efficiency in the system of Fig. 39 vs the pulse delay for a string of nine trapped ions with results for the individual ions (upper left) and the average over all ions (lower left). Right: CCD images of the fluorescence from the middle level $4P_{3/2}$ for the string of nine ions for various time delays between the $P$ and $S$ pulses. The absence of fluorescence signals high transfer efficiency. The maximum transfer occurs for delays between 2 and $3\mu \mathrm{s}$. Adapted from [484].

Figure 41

Upper left: Linkage pattern used in the experiment by [506]. Upper right: Pulse timing. The dressed state $|D⟩$ (or $|B⟩$) is prepared by half STIRAP with two microwave fields starting from sublevel $|+1⟩$. Then the amplitudes of the two microwave fields are held equal and constant, while the dressed qubit is driven by an rf field coupling the transitions $|0^{\prime }⟩\leftrightarrow |-1⟩$ and $|0^{\prime }⟩\leftrightarrow |+1⟩$. In the end, another half STIRAP is applied for the detection stage. Lower panel: Ramsey fringes verifying the coherence of the dressed qubit $\{|0^{\prime }⟩,|D⟩\}$. Adapted from [506].

Figure 42

Experimental demonstration of STIRAP in ${\mathrm{SO}}_2$ with ultraviolet transitions (227 nm for the pump field and 300 nm for the Stokes field), with pulse durations of 2.7 ns for the pump pulse and 3.1 ns for the Stokes pulse. Typical laser intensities were $10\mathrm{MW}/{\mathrm{cm}}^2$, yielding Rabi frequencies of about $10^{10}{\mathrm{s}}^{-1}$. The curves show the transfer efficiency vs the two-photon detuning with the Stokes laser on and off. Inset: Relevant energy-level scheme and linkage pattern. Adapted from [191].

Figure 43

Four-photon straddle STIRAP transfer to the rovibronic ground state $|v=0,J=0⟩$ and back in ultracold ${\mathrm{Cs}}_2$ molecules. Top: Transfer efficiency vs time (main frame) and vs detuning ${\mathrm{\Delta }}_4$ of the laser on the last transition (inset). Bottom: Temporal pulse shapes. From [95].

Figure 44

STIRAP scheme for forming RbCs molecules in their lowest rovibrational singlet ground state. The transfer from the Feshbach state $|i⟩$ to the rovibrational ground-state level $|v^{\prime \prime }=0,J^{\prime \prime }=0⟩$ involves the $|v^{\prime }=29⟩$ level belonging to the $b^3\mathrm{\Pi }(\mathrm{\Omega }=1)$ electronically excited state. Also shown are the wave functions that are coupled by the STIRAP lasers $P$ and $S$ with Rabi frequencies ${\mathrm{\Omega }}_p$ and ${\mathrm{\Omega }}_d$, respectively. From [494].

Figure 45

Formation of RbCs molecules in their lowest rovibrational singlet ground state by STIRAP. (a) Population histories of the Feshbach state. After the first STIRAP process, the Feshbach molecules disappear. A second STIRAP process drives the population from the lowest-lying molecular level back to the Feshbach state for detection. The one-way STIRAP efficiency is 90%. (b) Timing of the laser pulses. From [494].

Figure 46

STIRAP in ${}^{87}\mathrm{Rb}$ BEC: Transfer efficiency vs (a) two-photon detuning and (b) one-photon detuning. The dashed lines mark the (a) two-photon and (b) one-photon resonances. The dots are experimental data, the solid lines are numerical simulation, and the shaded (cyan) bands are numerical simulation with white magnetic-field noise included. From [125].

Figure 47

Spatial adiabatic passage of a particle between three potential wells. Tunneling induces couplings between the wells. Left: a linear string of wells, corresponding to a three-state $\mathrm{\Lambda }$ chain. Right: a two-dimensional set of wells corresponding to a loop system.

Figure 48

Pulse sequences for rotation gates with (a) two fractional-STIRAP processes, and (b) two STIRAP processes in the $\mathrm{\Lambda }$ system of Fig. 1. Adapted from [287] and [438]. (c) Pulse sequence for rotation gate created by two tripod-STIRAP processes, with the linkage pattern of Fig. 23. Adapted from [256]. (d) Pulse sequence for observation of a geometric phase without a dynamical phase in a tripod system. Adapted from [520].

Figure 49

Geometric phase gate demonstration. Left: Pulse sequence. Right: Populations $P_1$ (empty blue circles), $P_3$ (solid red circles), $P_4$ (magenta crosses), and $P_2$ (black asterisks). Adapted from [513].

Figure 50

Two atoms passing through the intersections of three cavities produce a five-state sequentially coupled chain of states. The axis of cavity 1 is spatially displaced with respect to the other axes, thereby producing counterintuitively timed atom-cavity couplings. From [294].

Figure 51

Top: Linkage pattern used for phase-coherent transfer of a spin wave between two atomic ensembles $A$ and $B$ (containing $N_A$ and $N_B$ atoms). The write process populates state $|G_A⟩$. $g\sqrt{N_A}$ and $g\sqrt{N_B}$ are the collective couplings of the magnons to the optical resonator mode, whereas ${\mathrm{\Omega }}_A$ and ${\mathrm{\Omega }}_B$ are the laser-atom couplings. The spin wave is transferred from $A$ to $B$ by a counterintuitive sequence of laser pulses (top left), with the pulse addressing $B$ preceding the pulse addressing $A$, thereby leaving the lossy photonic mode $|C⟩$ empty. Bottom: Phase coherence between the spin waves in the two ensembles after a partial transfer of a magnon. Adapted from [477].

Figure 52

(a) Parity oscillation of the two-ions Dicke state. (b) Oscillation of the squared spin of the half-excited Dicke state of four ions. From [369].

Figure 53

Demonstration of STIRAP in ${\mathrm{Pr}}^{3+}:{\mathrm{Y}}_2{\mathrm{SiO}}_5$. Top: Energy levels and linkages. Bottom: Population transfer efficiency vs peak Rabi frequency. From [262].

Figure 54

Demonstration of STIRAP in ${\mathrm{Pr}}^{3+}:{\mathrm{Y}}_2{\mathrm{SiO}}_5$. Top: Population transfer efficiency vs the two-photon detuning ($P$ frequency fixed, $S$ frequency varied), with a peak on two-photon resonance. Bottom: Population transfer efficiency vs pulse delay. The $P$ and $S$ fields are off resonance by the same detuning $\mathrm{\Delta }=2\pi \times 320\mathrm{kHz}$ (see Fig. 53), so that they are on two-photon resonance. The dashed curve shows the data with the incoherent population transfer contribution excluded. From [261].

Figure 55

STIRAP with $\mathrm{N}V$ centers in diamond. Left: relevant level scheme. The electronic spin sublevels $m_s=\pm 1$ are Zeeman split by 150 MHz, and the nuclear-spin sublevels $m_n$ by 2.2 MHz. State $m_s=0$ is coupled to states $m_s=-1$ or $+1$ by microwave pulses MW1 or MW2, respectively. Right: (a) pulse shapes and (b) pulse overlap for STIRAP transfer (left side) and two-photon Rabi oscillations (right side); (c) population of the target state in $m_s=+1$ monitored through fluorescence from state $E_y$ for $t_{\mathrm{rise}}=1.2\mu \mathrm{s}$, allowing adiabatic evolution; (d) same as in (c) but with $t_{\mathrm{rise}}=20\mathrm{ns}$, which is too steep to allow STIRAP transfer, and instead, two-photon Rabi oscillations are observed. Adapted from [171].

Figure 56

Voltage-driven superconducting Cooper-pair box (charge qubit, left), flux-driven loop (flux qubit, middle), and current-driven junction (phase qubit, right). From [594].

Figure 57

Experimental demonstration of STIRAP in a superconducting transmon circuit. Top (left: experiment, right: simulations): Contour plots of populations of state $|3⟩$ vs pulse delay and time. The white boxes bound the STIRAP region. Bottom: Contour plots of populations vs the single-photon detuning (vertical) and two-photon detuning (horizontal). The lower left frame shows the experimental data for the population of the ground state $|1⟩$. The other three frames show simulations for states $|1⟩$, $|2⟩$, and $|3⟩$. Adapted from [282].

Figure 58

Double-QD structure and carrier wave functions for the implementation of quantum gates by STIRAP. The confinement potential consists of two truncated-cone-shaped regions of low-band-gap material. (a), (b) The squared modulus of the hole ground and first excited wave function, respectively. (c) The charged exciton state, where the electron wave function (red, cut off for visibility) extends over the hole wave function due to the lighter electron mass and allows an optical coupling to both hole states. From [216].

Figure 59

Experimental demonstration of CPT in a single quantum dot. Top: Trion energy-level diagrams without (upper left) and with (upper right) magnetic field. $V$ ($H$) means vertical (horizontal) polarization. At zero magnetic field, the spin-flip Raman transitions are dipole forbidden, while they become allowed for nonzero magnetic field. The dashed lines isolate the three-state $\mathrm{\Lambda }$ linkage used in the experiment. Bottom: Experimental evidence of CPT in the probe absorption spectrum across transition $H1$ with the driving field applied on the transition $V2$. The dip reveals the formation of the dark state. Adapted from [581].

Figure 60

Experimental data [(a), (c)] and simulation [(b), (d)] of light propagating through a set of three WGs. (a), (b) Couplings that correspond to the counterintuitive interaction sequence, producing STIRAP-like transfer of light intensity from WG 1 to WG 3, while (c), (d) show couplings that correspond to the intuitive interaction sequence leading to Rabi oscillations. Horizontal red and blue lines added to (b) and (d) indicate the extent of $S$ and $P$ interactions. Adapted from [319].

Figure 61

Straddle-STIRAP analog in a WG structure for 3 (top) and 5 (bottom) middle WGs. Adapted from [98].

Figure 62

STIRAP-like action in WGs via a quasicontinuum. (a) WG arrangement: two WGs $W_1$ and $W_2$ couple to a common WG array $A$. (b) Experimental results, fluorescence image of light transfer by counterintuitive ordering of interactions: input enters in $W_1$ and transfers to $W_2$. (c) As in (b) but with intuitive ordering: input enters in $W_2$ and is lost into the quasicontinuum array $A$. Adapted from [118].

Figure 63

Top: Light polarization rotator based on piecewise adiabatic passage. The linear polarization vectors (solid lines) adiabatically follow the rotating fast polarization axis of the half-wave plates (dotted lines). Adapted from [110]. Bottom: STIRAP-inspired wireless energy transfer between three loops $L_1$, $L_2$, and $L_3$ carrying an ac current. The energy is transferred between loops $L_1$ and $L_3$, each of which undergoes a rotation of 90°. The middle loop is much larger (in order to ensure stronger couplings) and hence much lossier but it acquires only negligible energy. The relative orientations proceed from $L_1\perp L_2\parallel L_3$ to $L_1\parallel L_2\perp L_3$. Adapted from [194].

Figure 64

Schematics of three-mode optomechanical systems. (a) A mechanical mode coupled to two optical modes. (b) An optical mode coupled to two mechanical modes. Energy is transferred between optical cavity modes in (a) and the mechanical oscillators in (b). The optomechanical coupling is controlled by external laser fields (not shown) injected into the cavity with their frequency detuned from the cavity mode by the mechanical oscillators frequency. (a) Adapted from [115]. (b) Adapted from [116]. (c) The transmission of the weak probe field shows a minimum when the frequency of the control field is detuned from the cavity resonance (and thus the frequency of the probe field) by one mechanical resonance frequency. Adapted from [115].

Figure 65

A coherent superposition of $M=1$ and $-1$ states (leading to an alignment of the electron spin) in the $J=1$ level of the $H$ ${{}^3\mathrm{\Delta }}_1$ metastable electronic state of ThO is optically prepared either by optical pumping ([29]) or, for the next-generation experiments, by STIRAP ([383]). The spin precession angles ${\mathrm{\Phi }}_{+}$ and ${\mathrm{\Phi }}_{-}$ are measured for two opposite directions of the electric field. The difference $\mathrm{\Delta }\mathrm{\Phi }={\mathrm{\Phi }}_{+}-{\mathrm{\Phi }}_{-}$ yields, for known effective electric field $E_{\mathrm{eff}}$, the value of the electric dipole moment $d_e$ of the electron. From Cris Panda.

Figure 66

Top: The relevant level scheme of ThO for the eEDM measurement. The appropriate level in the $H$ state is populated either by optical pumping via the $A$ state or by STIRAP via the $C$ state. Adapted from [485]. Bottom: STIRAP transfer efficiency (left axis) from the ground state $X^1\mathrm{\Sigma }$ of ThO to the state $H$ ${{}^3\mathrm{\Delta }}_1$ relevant for the eEDM experiment. Positive values of the beam displacement correspond to the STIRAP arrangement. The right axis shows the enhancement of the population with respect to the optical-pumping approach. Adapted from [383].

Figure 67

Schematic potential energy curves as a function of a normal coordinate $q$. A state of well-defined parity is populated through STIRAP from the ground state. During subsequent evolution and because of parity-violating interaction, the state acquires a contribution from the other parity component, which is detected spectroscopically. Adapted from [411].

Figure 68

Left- and right-handed chirality. All couplings are the same except for a phase of $\varphi$ in the coupling between states 1 and 3.

Figure 69

Left: Level scheme for CaF. Microwave spectroscopy of $l=5$ Rydberg states requires efficient population of a state $m>40$, $l=4$. Traditional stepwise excitation is not possible because states such as the one with $n\approx 40$, $l=3$ decay rapidly. Pulsed excitation from the ground state to the $F^{\prime }$ state followed by optical-microwave STIRAP transfer allows bypassing the rapidly decaying state and depositing population into a sufficiently high-lying long-lived Rydberg state with $l=4$. Right: Schematics of a $l=0$ (core penetrating) and high $l$ (core nonpenetrating) orbital, with the CaF molecule chosen as an example.